In this paper we are concerned with diffeomorphometric approaches to image and
shape matching. To this end, we use time-dependent velocity fields \(u_t\)  occupying
some Hilbert space \(u_t \in V\) , where \(V\)  is continuously embedded in
\(\textsf {C}_0^k(\mathbb {R}^{d})\) , \(k\ge 1\)  inducing a curve \(\varphi _t\)  on a subgroup
\(\text{Diff}_V(\mathbb {R}^{d})\)  of diffeomorphisms
[1]}, [2]} via the following ordinary
differential equation
\(& \dot{\varphi _t} = u_t \circ \varphi _t\, , \qquad \varphi _0 = \text{id}\, .\) 
